Consecutive Numbers

It is known that the product of three consecutive natural numbers is always divisible by 6. Its means the product of $13 \times 19 \times n$ must be divisible by 6.

Putting $n = 6 \times k$

$13 \times 19 \times n \Rightarrow 13 \times 19 \times 6 \times k$

$\Rightarrow (13 \times 3) \times (19 \times 2) \times k$

$\Rightarrow 38 \times 39 \times k$

Now clearly $k$ can be replaced by either $37$ or $40$ . Since we are required to find the least value, we will replace $k$ by 37.

Hence the required value of $n$ will be

$n = 6 \times k = 6 \times 37 \Rightarrow \boxed{222}$